Optical patterning of trapped charge in nitrogen-doped diamond

The nitrogen-vacancy (NV) centre in diamond is emerging as a promising platform for solid-state quantum information processing and nanoscale metrology. Of interest in these applications is the manipulation of the NV charge, which can be attained by optical excitation. Here, we use two-colour optical microscopy to investigate the dynamics of NV photo-ionization, charge diffusion and trapping in type-1b diamond. We combine fixed-point laser excitation and scanning fluorescence imaging to locally alter the concentration of negatively charged NVs, and to subsequently probe the corresponding redistribution of charge. We uncover the formation of spatial patterns of trapped charge, which we qualitatively reproduce via a model of the interplay between photo-excited carriers and atomic defects. Further, by using the NV as a probe, we map the relative fraction of positively charged nitrogen on localized optical excitation. These observations may prove important to transporting quantum information between NVs or to developing three-dimensional, charge-based memories.

, we compare calculated and experimental (top central plot) NVpatterns upon 10 s of green light fixed-point excitation. Once again, we limit our analysis to three independent parameters, including the NVphotoionization rate at 532 nm, and the NV 0 and NVelectron and hole trapping rates, and , respectively. At the center of the multi-axis figure, all parameters take the values listed in Supplementary Table 1; in the peripheral images the parameter associated to each axis is altered to take the value listed on the lower left corner of each graph while all other parameters remain fixed. The scale bar (20 µm) and axes labels are the same for all images. . In all cases, the resulting pattern (not shown for brevity) is imaged via a red laser scan with center at the point of the laser park; the red laser power during the scanning is 2 mW, and the integration time per pixel is 1 ms. Numerical modeling is carried out using the conditions listed in Supplementary Table 1  ( ) Photoionization rate of NV -@632 nm @532 nm Hole production rate from NV 0 @632 nm @532 nm Electron capture rate by NV 0 kHz µm 3 [ †] Electron capture rate by N + 300 kHz µm 3 [10,11,14] Hole capture rate by NV -kHz µm 3 [ †] Hole capture rate by N 0 0.6 kHz µm 3 [10,11,14] Electron diffusion constant in diamond 5.5 µm 2 ns -1 [11] Hole diffusion constant in diamond 4.3 µm 2 ns -1 [11] NV density ppm [*], [13] nitrogen density ppm [*]

Initial conditions
Fraction of NVcenters after red-induced bleaching 0.07 Fraction of NVcenters after a green-laser scan 0.5

Supplementary Note 1 | Model of photoionization and charge diffusion in diamond
In this section, we provide an in-depth description of the theoretical model used to interpret and simulate our defect photoionization and charge diffusion observations. We begin by presenting the relevant defects in type-1b diamond and deriving the equations of their charge state dynamics. We next introduce carrier diffusion in diamond and combine the carrier diffusion equations with the equations of defect dynamics to form the complete model. We finally define the necessary initial and boundary conditions to solve the equations of the model.

i-Defects in type-1b diamond
The three most prevalent defects in type-1b diamond are the nitrogen-vacancy (NV), Under green (532 nm) or red (632 nm) illumination, NVcan be photoionised into NV 0 by ejecting an electron into the diamond conduction band via the successive absorption of two photons 3,4 . In this mechanism, the first photon excites NVto its optical excited state and the second photon ejects the electron into the conduction band within the lifetime of the optically excited state. Under green illumination, NV 0 can be photoionized back to NVby a similar mechanism involving the ejection of a hole into the diamond valence band and the successive absorption of two photons 3,4 (see Supplementary Fig. 2). The photoionization cross-sections of neither NV charge state are known precisely. Indicative photoionization rates for particular optical powers and focusing arrangements can be drawn from single center studies, which have measured the NVphotoionization rate to be ~5 Hz under ~1 μW (confocal) of red excitation and the NVand NV 0 photoionization rates to be ~25 Hz and ~60 Hz, respectively, under the same power of green excitation 4 .
The NV center may also trap charge carriers: NVis an attractively charged trap for holes and NV 0 is a neutral trap for electrons. Since both charge states are deep traps, their trapping mechanisms are likely to involve multi-phonon emission 5 (see Supplementary Fig. 2). However, the trapping cross-sections of the NV center have not yet been investigated and their values are unknown.
The N center is a deep donor in diamond and is known to exist in neutral (N 0 ) and positive (N + ) charge states. There is disagreement about the energy of the N donor level below the conduction band, with various photoabsorption and photoconductivity measurements reporting the energy in the broad range 1.7-2.2 eV. 6 -9 We therefore expect that N 0 will photoionize under either green or red excitation. However, we do not expect N + will photoionize back to N 0 by the ejection of a hole into the valence band under green or red excitation because this requires much higher energy photons (with energy greater than ~3.8 eV, see Supplementary suggest that there is negligible concentration of V 0 centers in our sample (no GR1 peak is observable at 741 nm in the fluorescence spectrum of Supplementary Fig. 1c) and so the V center will not be included in our model.
To confirm the transformation of NVinto NV 0 upon red illumination we alter our confocal microscope to separately collect the fluorescence from two complementary spectral windows. The first one, ranging from 605 nm to 615 nm is exclusively associated with NV 0 emission whereas the second one, from 637 nm to 850 nm, is mostly sensitive to NVfluorescence. The experiment of Supplementary Fig. 3 basically reproduces the protocol of Fig. 2 in the main text except that a weak green laserexciting both NVand NV 0is used to probe the resulting NV composition. The green laser intensity and exposure time is brought to a minimum so as to limit NV -NV 0 inter-conversion. As expected, photon collection from either spectral band leads to complementary NV signatures, with maximum NV 0 fluorescence when NVemission is minimum.

ii-Defect dynamics
The coupled dynamics of the charge state densities of NV and N centers as functions of position r and time t are described by the four master equations Furthermore, the second and third constraints in Eq. (3) can be used to reduce the four master equations to just the following two

iii-Carrier diffusion and trapping
Assuming drift-diffusion transport, the free electron and hole diffusion equations are where μm 2 /Vs and μm 2 /Vs are the approximately homogeneous and isotropic electron and hole mobilities in type-1b diamond 11 , respectively, μm 2 /ns and μm 2 /ns are the electron and hole diffusion constants at room temperature, respectively, is Boltzmann's constant, is the temperature.
where is now the Laplace operator in the two-dimensions of the observed photoionization and charge diffusion patterns.

Supplementary Note 2 | Numerical simulations
In this section, we provide the details of the simulations presented in the main text. As discussed above, the complete defect photoionization and charge diffusion model contains many parameters that are unknown and must be estimated. We begin by discussing our parameter estimations. We then describe the method of our simulations and our results for red and green excitations.
Note that due to the large number of parameters, we have not aimed to quantitatively simulate our observations, but to rather qualitatively demonstrate that we have identified, and our model describes, the main physical mechanisms at play. We leave it to future systematic measurements to precisely determine the parameters that quantify each mechanism.

i-Parameter estimation
The parameters to estimate are: , , , , , , , , and . As stated in the main text IR absorption spectroscopy indicates a concentration of neutral nitrogen impurities of ppm ( ) μm -3 , which here we assume as representative of the total nitrogen concentration . Based on the known correlation between the nitrogen and NV concentration in CVD diamond 13 , we set the total NV concentration at ppm or ~ μm -3 .
It is difficult to estimate beforehand the NV photo-ionization rates because prior studies have focused on single NVs where the laser power used (of order 1 µW) is much lower than in the present experiments 4 . The ionization rates grow quadratically in the limit of low laser powers but one anticipates a linear growth once the illumination intensity is sufficiently strong to saturate the NV first excited state. With these considerations in mind, a lower bound for the NV photoionization rate would be ~100 kHz for green light illumination; a much lower value is expected for red-light-induced photoionization. Given that photoconductivity measurements appear to imply that the N 0 photoionization rate under red excitation is much smaller than that under green excitation and is also small compared to NV -, we expect that under red isotropic and equal to . Using the measurements 10,11 μm 2 and μm 2 , we therefore expect that at room temperature kHz μm 3 and kHz μm 3 .
Since the NV center is a deeper trap than the N center, it is expected that .
Furthermore, since NVis an attractively charged trap for holes, whereas NV 0 is a neutral trap of electrons, one anticipates .
Our numerical modeling starts with the observations of Figs. 2, 3b (upper row), and 5 to determine all ionization rates (k -, k 0 , k N ) under 2 mW and 1.8 mW of red or green, fixed-point illumination, respectively. These values are subsequently used as a reference and correspondingly scaled for different illumination intensities. This scaling is made quadratic or linear to reflect the two-photon or one-photon processes governing NV and nitrogen ionization, respectively. We make use of this scaling to model the effect of the red laser scan (last step in the Sequence of Fig. 3a) on the green-induced NVpatterns (lower row in Fig. 3b) as well as the observed NVmaps on the split background (Fig. 4), see Supplementary Table 1.

ii-Red excitation
In principle, no free holes are created under red excitation because NV 0 cannot be directly photo-ionized. However, calculations based on an 'electron-only' model (i.e., ( ) ) are inadequate to reproduce our observations. An example is presented in Supplementary Fig. 4 where An 'electron-only' model, however, must be understood as an approximation because a variety of processes are likely to enable a low-rate back conversion from NV 0 to NV -. As explained in the main text, even a small hole production rate has a significant effect given the very low hole-trapping efficiency of nitrogen (Supplementary Table 1). Good agreement is attained with ⁄ (center image in Supplementary Fig. 4); note that higher ratios lead to unobserved behaviour (upper image in Supplementary Fig. 4).
The nature of the process at play is difficult to identify at present: Comparison of similarly-sized patterns from different combinations of laser power and exposure time (e.g., 2 mW for ~10 s versus 1 mW for ~25 s in Supplementary Fig. 6a) reveals comparable NVconcentrations both in the bright and dark sections of the resulting ring-like structures. We surmise, therefore, that the electron and hole production rates scale similarly with light intensity suggesting that hole-production under red excitation is also the result of a two-photon process (at least within the range of laser intensities investigated in this study). Additional work, however, will be required to ascertain whether this process originates from NV 0 back-conversion to NV -(as modeled herein) or from dynamic charge inter-conversion of some other, lower-concentration defect in our diamond crystal.
While a comprehensive description of the effect of each parameter on the simulated pattern is impractical, additional insight on the model accuracy can be attained by examining the model sensitivity to two key variables, namely, the electron and hole trapping rates. For example, a brighter-than-background torus is generated when is made ten-fold larger (or when is made ten-fold smaller) than the value listed in Supplementary Table 1 (central   image). In the opposite regime, a fully bleached disk results when ( ) is made ten-fold smaller (larger) than the optimum, a consequence of the enhanced capture of holes by the NVs.
Note that these non-optimum values also produce patterns extending over an area larger (or smaller) than that observed.
Although our modeled patterns are sensitive to relatively small (e.g., 20%) changes of individual parameters, we find a considerably broader error margin if several parameters are modified simultaneously. For example, we find moderate agreement with experiment using alternative parameter sets where the ratios between all ionizing rates are kept constant. These alternate 'optimal' sets, however, are not dramatically different from each other, and typically correspond to correlated parameter changes by a factor 2 to 4. The error margins listed in Supplementary Table 1 reflect the range within parameter space containing these alternate 'optimal' sets. We emphasize, however, that best overall agreement is attained with the 'main' parameter set as listed.
While throughout our modeling we kept constant the nitrogen and NV concentrations, a question of interest is how local, mesoscale changesby as much as a factor ~2 over distances of a few micronsinfluence the charge dynamics. In general, we find that separately changing or leads to the appearance of pattern features not observed experimentally, even if a new "optimal" set of parameters is identified. The nitrogen ionization rate proved to be particularly sensitive to these changes, which can be qualitatively understood from the key role   Supplementary Fig. 5). As discussed in the main text, the listed set of parameters also yields pattern growth rates in qualitative agreement with the experiment, both for red (Fig.   2g) and green (Fig. 5h) illumination (see also Supplementary Fig. 6)

iv-Drift versus diffusion versus trapping
To investigate the influence of electrostatic fields on the dynamics of photo-ionized carriers, we start by considering the quasi-steady state where the fluorescence pattern is only changing near the expanding outer radius. Limiting for now our analysis to red excitation, the radial charge distribution in two dimensions is crudely given by, where and respectively represent the inner and outer radii in the red-induced pattern (see Fig. 2), and we assume that the inner region under optical excitation has gained a uniform positive charge per unit length . We also assume that the fluorescent ring features an equally uniform (but negative) charge distribution so that the total net charge is null. The radial electric field produced by this charge distribution is At we thus find the minimum value | | . Setting to 5 nsthe time needed for the electron to diffuse away from the inner ringand defining as the electronic charge, we conclude that for | | to drop below 1 the normalized charge per unit length ⁄ must be greater than 10 6 µm -3 . Given the nitrogen concentration ppm and considering an optical spot with transverse dimensions of µm 2 we conclude carrier transport becomes driftdominated when the nitrogen ionization approximately equals or exceeds 10%. As seen in Fig.   6d in the main text, the concentration of positively charged nitrogen does not exceed 0.4 ppm, suggesting that electrostatic fields do not have a major impact on the carrier transport dynamics under the present conditions.
To gain a more quantitative understanding we explicitly calculate the net charge density and electrostatic field resulting from 10 s of red (and green) illumination ( Supplementary Fig. 7).
Here we follow a self-consistent approach were the drift force is initially neglected from the master equations. As expected, we find that the electrostatic field reaches its maximum within the inner section of the fluorescence ring ( Supplementary Fig. 7c). The electric field is substantially stronger for green illumination, a consequence of the relatively more efficient ionization of nitrogen at this wavelength. The results also show that the magnitude of the diffusion/drift ratio | ( )| consistently takes values larger than unity throughout the pattern, thus confirming the dominance of diffusion under the present experimental conditions. It is worth mentioning that unlike the estimate for | | above, this time no assumption is made on the spatial distribution of free electrons, which results from the (drift-free) master equations.
Finally, we gain further insight in the hierarchy of terms in Supplementary Eq. (7) by separately calculating diffusion and capture terms. The results are presented in Supplementary   Fig. (7d) for 10 s of red (left) or green (right) illumination. In both cases, we find that trapping terms are dominant, consistent with the notion of a 'reaction-diffusion' regime where the pattern boundary progresses once an equilibrium concentration of charged defects is reached (see main text).